Legendre modified moments for Euler s constant

Transcription

1 Journal of Computational and Applied Mathematics 29 (28) Legendre modified moments for Euler s constant Marc Prévost Laboratoire de Mathématiques Pures et Appliquées, Joseph Liouville, Université du Littoral, Côte d Opale, Centre Universitaire de la Mi-Voix, Bâtiment H. Poincaré, 5 rue F. Buisson, BP 699, Calais Cedex, France Received 22 November 26; received in revised form 2 September 27 Dedicated to Claude Brezinsi at the occasion of his retirement. Abstract Polynomial moments are often used for the computation of Gauss quadrature to stabilize the numerical calculation of the orthogonal polynomials, see [W. Gautschi, Computational aspects of orthogonal polynomials, in: P. Nevai (Ed.), Orthogonal Polynomials-Theory and Practice, NATO ASI Series, Series C: Mathematical and Physical Sciences, vol Kluwer, Dordrecht, 99, pp [6]; W. Gautschi, On the sensitivity of orthogonal polynomials to perturbations in the moments, Numer. Math. 48(4) (986) [5]; W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3(3) (982) [4]] or numerical resolution of linear systems [C. Brezinsi, Padé-type approximation and general orthogonal polynomials, ISNM, vol. 5, Basel, Boston, Stuttgart, Birhäuser, 98 [3]]. These modified moments can also be used to accelerate the convergence of sequences to a real or complex numbers if the error satisfies some properties as done in [C. Brezinsi, Accélération de la convergence en analyse numérique, Lecture Notes in Mathematics, vol Springer, Berlin, New Yor, 977; M. Prévost, Padé-type approximants with orthogonal generating polynomials, J. Comput. Appl. Math. 9(4) (983) ]. In this paper, we use Legendre modified moments to accelerate the convergence of the sequence H n log(n + ) to the Euler s constant γ. A formula for the error is given. It is proved that it is a totally monotonic sequence. At last, we give applications to the arithmetic property of γ. 27 Elsevier B.V. All rights reserved. MSC: 4A2; 65B Keywords: Legendre moments; Euler s constant; Padé approximations. Introduction The Euler constant γ is the limit of the sequence H n log(n + ), where H n is the harmonic number defined by. address: prevost@lmpa.univ-littoral.fr /$ - see front matter 27 Elsevier B.V. All rights reserved. doi:.6/j.cam

2 M. Prévost / Journal of Computational and Applied Mathematics 29 (28) An integral representation for Euler s constant is ( γ ln u + ) du. () u The sequence (S n ) ( u n S n : log u + ) un du u H n log(n + ) converges to γ but very slowly, as O(/n). The error γ S n satisfies ( γ S n u n log u + ) du (2) u and can be considered as moments with respect to the weight function w(u) (/ log u + /( u)) on the interval [, ]. 2. Legendre modified moments Suppose we are given a sequence of real or complex numbers (x n ) n converging to l and satisfying the property x n l u n dμ(u) where dμ is a positive measure on the interval [, ]. If the error of a sequence is of this form, then a way to decrease this error and so accelerate the convergence is to use modified moments, i.e. by replacing the monomial u n by some suitable polynomials P n normalized by P n (). For P n (u) α (n) u, P n(u) dμ(u) n α (n) u dμ(u) n α (n) limit l and the transformed sequence (y n ) defined by y n : n α (n) y n l P n (u) dμ(u). (x l) n α (n) x l. In that case, the error between the x becomes To improve the convergence, the polynomial P n can be chosen to be orthogonal with respect to some weight. In our case, because of the behavior of the weight function around (w(u) O(), u ), a good choice will be the shifted Legendre polynomials which are orthogonal on [, ] Pn (t)p m (t) dt, n m. These polynomials can be expressed in different bases P n (t) ( ) 2 n t n (t ) (3) n n + ( ) n+ t. (4)

3 486 M. Prévost / Journal of Computational and Applied Mathematics 29 (28) Theorem. For m n, let us define ( J n,m : u n m Pn (u) ln u + ) du, (5) u ( u n m Pn L n,m : (u) ) du, (6) ln u ( u n m Pn A n,m : (u) ) du (7) u then γ A n,m L n,m + J n,m, (8) n n + L n,m ( ) n+ ln(n m + + ), (9) A n,m 2H n. () Proof. We first prove the identity (8) lining Euler s constant γ, the linear combination of logarithms numbers L n,m, the rational numbers A n,m and the integrals J n,m. From formula (), one substitutes the integrand w(u)(/ ln u+/( u)) by an approximation involving Legendre Polynomials as follows: ( γ ln u + ) du u ( u n m Pn (u) + un m Pn (u) ) du + u n m Pn (u)w(u) du. ln u u The expression (4) of Pn leads to analogous expressions L n,m. By linearity n n + ( u L n,m ( ) n+ +n m ) du () ln u n n + ( ) n+ ln(n m + + ). (2) A n,m is treated quite differently: from the orthogonality relation between two polynomials P n Pn (u)q(u) du for all polynomial q of degree less than n (3) by taing q(u) ( u n m )/( u), another expression for A n,m is A n,m P n (u)q(u) du + P n (u) u and so A n,m is independent of m n. Let us now compute the integral in (4). Legendre polynomials satisfy a three term recurrence relation which is Pn du () P n (u) du (4) u (n + )Pn+ (u) (2n + )(2u )P n (u) np n (u), (5) P (u), P (u) 2u. (6)

4 M. Prévost / Journal of Computational and Applied Mathematics 29 (28) Thus, A n,m s also satisfy a similar recurrence relation (n + )A n+,m (2n + )A n,m na n,m, (7) A,m, A,m 2. (8) With (7) and (8), it is not difficult to prove that A n,m 2H n, m n. (9) In Section 4,wewill prove the asymptotic formula γ A n,m L n,m + O(4 n ). (2) Actually, we will prove that the error term J n,m is a totally monotone sequence (i.e. a sequence of monomial moments with respect a positive measure), converging to as 4 n. Before, we need to investigate the analytic property of the weight function w. 3. An interesting integral representation for Euler s constant Euler constant can be written as sum of series or with integral representation. (See Constants/Gamma/gamma.html). A complete study (with more than 3 references) can be found in [7]. In this section, we give an integral representation of the weight w in Lemma 2, which leads to a formula for γ first proved by Schlömlich [2] in 88 and rediscovered by Krämer [7, p. 29]. We give another proof of this formula. Lemma 2. The function / ln( u) + /u is a Marov Stieltjes function. More precisely, ln( u) + u μ(t) dt, (2) ut where the weight function μ is μ(t) : t(ln 2 (/t ) + π 2 ). Proof. After a change of variable (u ( u) and x /t ), formula (2) is equivalent to ln(u) + u x + u ln 2 dx. (22) x + π2 The weight function μ can be found with the Stieltjes inversion formula (see [6]). Another way to prove formula (22) is to apply residue theorem to the function f(x): x + u ln x + iπ. Taing the determination of ln x on the complex plane cut along the positive real axis, the poles of f are x u and x. Let us define γ r a small semi-circle z re iθ, π/2 θ π/2,r>. D r + the line z x + ir, x running from to R, Γ R the circle z Re iθ, θ 2π and Dr the line z x ir, for x from R to. Now, we compute C f(x)dx where C is the union of D+ r, Γ R, Dr and γ r, with the theorem of residue to obtain x + u ( ln x + iπ + ln x iπ ) ( ) 2iπ dx x + u ln 2 dx (23) ( x + π 2 2iπ ln u + ). (24) u Now, we are in position to prove the integral of Schlömlich for Euler s constant γ with a new method.

5 488 M. Prévost / Journal of Computational and Applied Mathematics 29 (28) Theorem 3. The Euler s constant γ satisfies + ln( + e z )e z γ z 2 + π 2 dz (25) Proof. In the integral representation () of γ, let us substitute the integrand by the expression (2). This leads to γ ln( t) t t(ln 2 dt (/t ) + π 2 ) (26) ln( + e z )e z z 2 + π 2 dz (27) with the change of variable t ( + e z ). With the expression of the weight function as a Marov Stieltjes function, it is possible to deduce some interesting properties for the error, for example to show that it is a sequence of moments. 4. Behavior of the error Theorem 4. For each fixed integer m, the sequence (( ) m J n,m ) n defined in Theorem is totally monotonic. More precisely ( ) m J n,m /4 where the weight function is ρ m (v) (+ 4v)/2 ( 4v)/2 v n ρ m (v) dv O(4 n ), (28) ( u u 2 ) m v ( )) du. uv (u u 2 v) (π 2 + ln 2 uv u 2 u + v Proof. J n,m un m Pn (u)(/ ln u + /( u)) du appears as Legendre modified moments of the weight function (/ ln u + /( u)). For some particular cases of weight function, a sequence of polynomial modified moments can be itself a sequence of monomial moments, with respect to a positive measure (see []). Using Rodrigues formula for orthogonal polynomials, Lemma 2, Fubini s theorem and after n integrations by parts, it arises J n,m n m ( )n u n! n m ( )n u n! ( ) n n! d n ( du n (un ( u) n ) ln u + ) du (29) u d n du n (un ( u) n ) du μ(t) dt (3) ( u)t ( u n ( u) n du dn u n m ) du n μ(t) dt. (3) ( u)t The computation of d n /du n (u n m /( ( u)t)) needs the partial decomposition of the rational function u n m /( ( u)t) q(u) + ((t )/t) n m /( ( u)t), where q is polynomial of degree n m. Another expression of J n,m is then J n,m ( ) t n m u n ( u) n t n μ(t) dt du. (32) t n+ ( ( u)t)

6 We do the following change of variable v u( u)( t) ( u)t M. Prévost / Journal of Computational and Applied Mathematics 29 (28) [, /4] t (v) u2 u + v [, ]. (u v)(u ) Let (v) and 2 (v) denote the two roots of the quadratic equation v u u 2, (v) ( + 4v)/2, 2 (v) ( 4v)/2. /4 2 J n,m v n (v) ( ) (v) m u 2 μ( (v)) ( ) m du dv (v) (v) (u )(u v) 2 ( u) (v) which can be simplified to give the result. (33) As quoted in [2,9,], the sequence (A n,m L n,m ) n converging to γ with an error totally monotonic on the interval [,R] with R 4 (2/R can be accelerated by the ε-algorithm to obtain an error of order O(((2/R ) ) 2 ) n ) O((7 48) n ). Remar. There exists some sequence converging to γ with an error of order O(e 8n ) [], but they do not provide arithmetic property for γ, as we can do in the last section. 5. Approximation of γ by rational numbers In the numerical computation of formula n n + L n,m ( ) n+ ln(n m + + ), the problem is the evaluation of logarithmic functions. A mean to avoid this drawbac is the substitution of ln(n m + + ) by some suitable approximations. We will show now that Padé approximants are good enough to preserve the speed of convergence O(4 n ). Another expression of L n,m is L n,m ln(n m + ) + ( n )( ) ( n + ( ) n+ ln + ). (34) n m + By substituting in L n,m,ln(n + + m) by its Padé approximant [n/n], a rational approximation is obtained as following: for n m + 2 p,p Z, let us define n n + L n,m : p[n/n] t + ( ) n+ [n/n] t/(n m+), (35) where [n/n] is the Padé approximant of ln( + t) at t t ( ) n n n + t i +n ( ) i i i + [n/n] t. (36) n n n + t n If we replace in (8), the quantities L n,m by L n,m, we get an approximation of γ by rational numbers: Theorem 5. For integers n, m, p such that n m + 2 p γ A n,m L n,m + O(4 n ). (37)

7 49 M. Prévost / Journal of Computational and Applied Mathematics 29 (28) Proof. The Padé error for the logarithmic function is ln( + x) [n/n] x ( )n x n+ t n ( t) n Pn ( /x) dt. (38) n+ ( + xt) Let us set L n n + n,m : ln(n m + ) + ( ) n+ [n/n] t/(n m+). (39) We have to evaluate the difference δ n,m : L n,m L n,m. For sae of simplicity, we set ζ /(n m + ). n n + δ n,m ( ) n+ (ln( + ζ ) [n/n] tζ ) (4) n n + ( ) ζ n+ t n ( t) n Pn ( ζ ) dt. (4) n+ ( + ζ t) Since ζ [, ] and Pn has all its roots in [, ], ζn /P n ( ζ ) / P n ( ). On the other hand, the integral t n ( t) n dt 4 n dt 4 n. (42) n+ n+ ( + ζ t) ( + ζ t) nζ So, n n + ζ n+ t n ( t) n δ n,m P n ( ζ ) dt n+ ( + ζ t) n n + ζ P n ( ) 4 n nζ n n + np n ( ) 4 n np P n ( ) 4 n n ( ) (n4n ). The goal is partly reached since the error between L n,m and its approximation is less than J n,m. Now, let us consider the approximation of ln(n m + ). It is difficult to approximate this number (which tends to infinity) with an error less than 4 n. So, we consider sequences of integers n, such that n m + is a power of 2: n m + 2 p. With this hypothesis, ln(n m + ) p ln 2. In (38), if x, ln 2 [n/n] x ( ) n /Pn ( ) (tn ( t) n /( + t) n+ ) dt. The asymptotic for Legendre polynomials are well nown P n (α) (α + α 2 ) n for α R\[, ]. Thus the shifted Legendre Polynomials satisfy P n (t) ((2t ) + 2 t 2 t) n for t R\[, ]. The maximum of the fraction (t( t)/( + t)) for t [, ] is obtained for t 2, and its value is (3 2 2). Thus ln 2 [n/n] x (3 2 2) n (3 + 2 ln 2. (43) n 2) For n m + 2 p,ln(2 p ) p[n/n] x p(3 2 2) 2n which is a o(4 n /n). At last, the error L n,m L n,m satisfies L n,m L n,m O(4 n /n).

8 M. Prévost / Journal of Computational and Applied Mathematics 29 (28) Now, γ A n,m + L n,m J n,m + L n,m L n,m O(4 n ) + O(4 n /n) O(J n,m ) (44) and the theorem is proved. 6. Application to the arithmetic property of γ The irrationality of γ remains an open question. In this section, we prove some sufficient conditions for the proof of the irrationality of Euler s constant. Corollary 6 is similar to the results found in [3,4,8], but it involves only the computation of rational numbers. On the other hand, Corollary 7 is new because it depends on the decrease of a numerical sequence involving logarithms numbers. Corollary 6. Let us denote by {x} the fractional part of the real number x: {x} :x x and d n : LCM(,...,n) (lower common divisor). If for some integer m, {d 2 p +m ( ) m L 2 p +m,m} does not converge to when p tends to infinity, then γ is irrational. Proof. Suppose that γ is rational, then there exists a pair of integers A, B such that γ A/B. Then, for n greater than some integer N, d n γ is an integer. For integers n, m, p such that n m + 2 p, the relation leads to γ A n,m L n,m + O(J n,m ), d n γ d n A n,m d n L n,m + d n O(J n,m ). A n,m 2H n,sod n A n,m is an integer, and thus the fractional part of ( ) m d n L n,m is equal to the fractional part of the positive sequence ( ) m d n O(J n,m ) which converges to zero since lim n dn /n e []. So, if for some integer m, the fractional part ({d 2 p +m ( ) m L 2 p +m,m}) p does not converge to, then γ is irrational. Another sufficient condition comes from the property of the error term in the asymptotic formula (2) and from the upper and lower bound of the LCM(,...,n): Corollary 7. Let P be the following property: A sequence (x n ) n satisfies P if N N, n N, x n x n+ <. If for some integer m, the sequence {d 2 p( ) m L 2 p,m} satisfies P then γ is irrational. Proof. For the proof, we exploit the property of totally monotonic sequences (TMS). A sequence u n is called TMS if there exists a non negative measure dμ with infinitely many points of increase such that n N, u n x n dμ(x). If the support of the measure dμ is the interval [, /R], then n, u n+ /u n R and lim n u n+ /u n R. IfR, it is equivalent to n N, N, ( ) Δ (u n )>, where Δ (u n ) : u n and Δ + u n Δ u n+ Δ u n. (See [6, p. 8].)

9 492 M. Prévost / Journal of Computational and Applied Mathematics 29 (28) The previous properties can be applied to the sequence J n,m for which we prove some convergence properties. Since {d n ( ) m J n,m }d n ( ) m J n,m {d n ( ) m L n,m }, if they are not satisfied by {d n ( ) m L n,m } then γ is irrational. First we will prove that J n,m satisfies d 2n ( ) m J 2n,m <d n ( ) m J n,m : the numbers d n and J n,m satisfy 2 n d n < e.39n (see [5, pp. 2 3] for the lower bound and [] for the upper one) J n+,m /J n,m < 4 (property of totally monotonic sequence [6, p. 35]). d n J n,m 2 n > d 2n J 2n,m e.39 2n 4n >.4. Thus, for all integer m, (d 2 p( ) m J 2 p,m) p N is a positive decreasing sequence, converging to. So, if ({d 2 p( ) m L 2 p,m}) p is nondecreasing for p greater than any integer, then γ is irrational. Consequence: If for some m, {d 2 p( ) m L 2 p,m} satisfies P, then γ is irrational. Suppose that for some m, and some p {d 2 p( ) m L 2 p,m} < {d 2 p+( ) m L 2 p+,m}, then it implies that if γ is rational then its denominator is greater than 2 p. Numerical computation show that it is true for m,p 5. So, if γ is rational, then its denominator is greater than Acnowledgment I would than my colleague S. Eliahou for Ref. [8]. References [] R.P. Brent, E.M. McMillan, Some new algorithms for high-precision computation of Euler s constant, Math. Comput. 34 (98) [2] C. Brezinsi, Accélération de la convergence en analyse numérique, Lecture Notes in Mathematics, vol Springer, Berlin, New Yor, 977. [3] C. Brezinsi, Padé-type approximation and general orthogonal polynomials, ISNM, vol. 5, Birhäuser, Basel, Boston, Stuttgart, 98. [4] W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3 (3) (982) [5] W. Gautschi, On the sensitivity of orthogonal polynomials to perturbations in the moments, Numer. Math. 48 (4) (986) [6] W. Gautschi, Computational aspects of orthogonal polynomials, in: P. Nevai (Ed.), Orthogonal Polynomials Theory and Practice, NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 294, Kluwer, Dordrecht, 99, pp [7] S. Krämer, Die Eulersche Konstante γ und verwandte Zahlen, Thesis, Mathematische Institut der Georg-August-Universität Göttingen, 27. [8] T.H. Pilehrood, Kh.H. Pilehrood, Criteria for irrationality of generalized Euler s constant, J. Number Theory 8 (24) [9] M. Prévost, Padé-type approximants with orthogonal generating polynomials, J. Comput. Appl. Math. 9 (4) (983) [] M. Prévost, Acceleration of some logarithmic sequences, J. Comput. Appl. Math. 55 (3) (994) [] J. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (962) [2] O. Schlömlich, Zu Schröder s Infinit. Werth von (u) n du. Z. Math. Phys. angew. Math. ZDB (88), S.7 9. JFM [3] J. Sondow, Criteria for irrationality of Euler s constant, Proc. Amer. Math. Soc. 3 (23) [4] J. Sondow, W. Zudilin, Euler s constant, q-logarithms, and formulas of Ramanujan and Gosper, Ramanujan J. 2 (2) (26) [5] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Institut Elie Cartan, Université de Nancy, 99. [6] D.V. Widder, The Laplace Transform, Princeton Mathematical Series, Princeton University Press, NJ, 94.